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Cours scientifiques - FMA_2S007_EP : Series of Functions, Differential Equations

Domaine > Mathématiques.

Descriptif

Series of functions, differential equations (FMA_2S007_EP) builds upon the topology notions studied in Topology and multivariable calculus (FMA_2F002_EP) to allow for a more profound study of functions. Examining functions as limits of simpler ones (e.g. for approximation problems) is made possible in a rigorous manner thanks to topological ideas. This provides the possibility of using crucial tools in many scientific fields; the most striking one being Fourier series (first designed to solve the heat equation and now ubiquitous in science and, in a hidden manner, in daily life). The second part of the course deals with a wide array of differential equations, permitting students to better understand complex physical questions.

Objectifs pédagogiques

At the end of this course, students should feel comfortable with the vairous notions of convergence of sequences and series of functions, especially power series and Fourier series. They will also know the basics of the analysis of ordinary differential equations.

Pour les étudiants du diplôme Programmes d'échange internationaux

Vous devez avoir validé l'équation suivante : UE FMA_2F002_EP

One variable and multivariable calculus (differentiation, integration, numerical series), fundamentals of topology of normed vector spaces as seen in any first or second year undergraduate class (such as FMA_2F002-EP).

Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vous devez avoir validé l'équation suivante : UE FMA_2F002_EP

One variable and multivariable calculus (differentiation, integration, numerical series), fundamentals of topology of normed vector spaces as seen in any first or second year undergraduate class (such as FMA_2F002-EP).

Format des notes

Numérique sur 20

Littérale/grade américain

Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vos modalités d'acquisition :

2 homework problems (HW1-2), 1 Mid-term exam (MT), 1 final exam (FE). Final grade: 0.25*(HW1+HW2)+0.5*max(FE,0.5*(FE+MT))
 
Mid-term and final exams are open book.
 

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)
    L'UE est acquise si Note finale >= 10
    • Crédits ECTS acquis : 5 ECTS

    La note obtenue rentre dans le calcul de votre GPA.

    Pour les étudiants du diplôme Programmes d'échange internationaux

    Vos modalités d'acquisition :

    2 homework problems (HW1-2), 1 Mid-term exam (MT), 1 final exam (FE). Final grade: 0.25*(HW1+HW2)+0.5*max(FE,0.5*(FE+MT))
     
    Mid-term and final exams are open book.
     

    Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)

      Programme détaillé

      I. Pointwise vs. uniform convergence of sequences and series of functions

      II. Power series

      III. Fourier series

      IV. Ordinary differential equations 

      Mots clés

      Pointwise convergence - Uniform convergence - Power series - Fourier series - Differential equations

      Méthodes pédagogiques

      1.5-hour blackboard lectures based on lecture notes with 2-hour exercise sessions per week
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